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Ellipse Involute

\begin{figure}\begin{center}\BoxedEPSF{EllipseInvolute.epsf scaled 800}\end{center}\end{figure}

From Ellipse, the Tangent Vector is

{\bf T}=\left[{\matrix{-a\sin t\cr b\cos t}}\right],
\end{displaymath} (1)

and the Arc Length is
s=a\int\sqrt{1-e^2\sin^2 t}\,dt = aE(t,e),
\end{displaymath} (2)

where $E(t,e)$ is an incomplete Elliptic Integral of the Second Kind. Therefore,
$\displaystyle {\bf r}_i$ $\textstyle =$ $\displaystyle {\bf r}-s\hat{\bf T}=\left[\begin{array}{c}a\cos t\\  b\sin t\end{array}\right]-aeE(t,e)\left[\begin{array}{c}-a\sin t\\  b\cos t\end{array}\right]$ (3)
  $\textstyle =$ $\displaystyle \left[\begin{array}{c}a\{\cos t+aeE(t,e)\sin t\}\\  b\{\sin t-aeE(t,e)\cos t\}.\end{array}\right]$ (4)

© 1996-9 Eric W. Weisstein